None of the four conditions is spare

The rung below states four conditions and reports that exactly one sharing rule satisfies all of them. That is half a theorem.

A characterisation has two halves and the second is the one that decides whether the list of conditions was well chosen. Sufficiency says the four conditions leave only one rule. Independence says none of the four could have been left out — that dropping any single one admits something else. Without the second, a list of four conditions might really be a list of three with a passenger.

What the game has to provide #

The figure’s game is chosen rather than convenient, and the reason is that two of the four conditions are vacuous on most games.

Alike is treated alike says nothing unless two players really are interchangeable — unless swapping them leaves every coalition’s value unchanged. No pay for no work says nothing unless some player really adds nothing anywhere. On a game where neither holds, every rule satisfies both conditions by default and the figure argues nothing.

So the game needs both, and with three players it cannot have both usefully: a null player plus two interchangeable ones leaves two real players who are identical, and then every rule drawn gives the same answer. Four players is the smallest that works — two alike, one worth more, one adding nothing — and the figure checks all three of those properties on its own table before drawing anything.

A demonstration that a condition is not spare needs a case where the condition bites. That is obvious once stated and it is the sort of requirement that is silently violated by a figure built from whatever game was already in the file.

The four rules #

Each column holds a rule satisfying three conditions and breaking one. All four are computed on the same game, and all four are asserted to give a different answer from the average over orders — which is what dropping a condition admits something else means operationally.

Without efficiency: half of what each adds. Take the Shapley value and halve it. Every player who is alike still gets the same, every null player still gets nothing, and adding two games still adds the shares — because halving is linear. What fails is that the shares no longer total what the group earns.

Without symmetry: what each adds, in one fixed order. Line the players up in a stated order and give each what they add on arriving. The shares total the whole, by telescoping; a null player adds nothing wherever they stand, so gets nothing; and marginal contributions are linear in the game, so the rule is additive. What fails is that two interchangeable players get different amounts, because one of them arrived first.

Without the null-player condition: split it equally. Everybody gets the same share of the total. Efficient, symmetric, additive — and a player who adds nothing anywhere still collects.

Without additivity: split it in proportion to what each earns alone. Efficient by construction, symmetric because interchangeable players earn the same alone, and a null player earns nothing alone so gets nothing. What fails is additivity: proportions of a sum are not the sum of proportions, and the figure checks that on a stated pair of games rather than arguing it.

Why the marginal-vector rule is the interesting one #

Three of the four columns hold rules nobody would propose. The symmetry column holds one that is proposed constantly, under a different description, and it is worth the paragraph.

Giving each player what they add on arriving in a fixed order is exactly one term of the average the Shapley value takes. It is called a marginal vector, there is one for each of the $n!$ orders, and the Shapley value is their average.

So the rule breaking symmetry is not an artificial construction: it is what happens when the averaging step is skipped. And every marginal vector satisfies efficiency, the null-player condition and additivity — so the whole content of the symmetry condition is the instruction to average, and without it any single order will do.

That reading makes the theorem read differently. The four conditions are not four independent demands; three of them are satisfied by every marginal vector, and the fourth says to treat the orders alike. The rule is the symmetric combination of the obvious rules, and the uniqueness theorem is the statement that averaging is the only symmetric combination that stays efficient and additive.

The marginal vector, written out #

The symmetry column deserves its own look, because the rule in it is the Shapley value with one step removed and seeing that step removed is the clearest view of what the average is doing.

Six rows for three players, twenty-four for four, and $n!$ in general. Each row divides the whole exactly; each gives a null player nothing; each is linear in the game. What no single row does is treat interchangeable players alike, because a row is an order and an order distinguishes them.

So the four conditions decompose cleanly:

• efficiency, the null-player condition and additivity are satisfied by every row;
• symmetry is what forces a combination of the rows that treats them alike;
• and among the combinations, efficiency and additivity force it to be an average with equal weights.

The theorem is that averaging the rows is the only way to symmetrise them. Any other symmetric combination — the median of the rows, say — fails additivity, and any other weighting fails symmetry. That is why the figure of six rows is not merely an illustration of the definition; it is the object the four conditions select from.

Where the same method gives an impossibility #

The axiomatic method is a machine with two outputs, and this field contains the other one.

Here, four conditions on a sharing rule leave exactly one rule. In four conditions and no rule, four conditions on a voting rule leave none — Arrow’s theorem, where every rule breaks one and the exhaustion that shows it is over every profile of ballots.

The two results have the same shape and opposite verdicts, and the difference is worth locating. Arrow’s conditions include independence of irrelevant alternatives, which demands that the ranking of two candidates depend only on how voters rank those two. That is a very strong requirement — it forbids the rule from consulting anything about intensity or about the rest of the ballot — and it is what makes the list unsatisfiable.

The method does not decide which output it will produce, and that is its main virtue. A list of conditions is a specification, the theorem says what meets it, and nothing is as informative an answer as exactly one. A field that only ever proved uniqueness would be a field whose conditions were chosen to be satisfiable.

Which condition would be dropped in practice #

Independence says all four are needed for this rule. It does not say all four are wanted, and each of the four has a literature of dropping it deliberately.

Symmetry is dropped most often. Players may differ in ways the game’s numbers do not record — seniority, bargaining strength, prior investment — and the weighted Shapley values take a weighted average over orders instead of a flat one, which is the same move a lottery over assignments makes with the weights chosen rather than uniform. They satisfy the other three conditions and form a family, one member per assignment of weights, with the Shapley value the equal-weights case.

Additivity is dropped next. It is the least intuitive of the four and the one doing the most work, and rules failing it — the nucleolus, the proportional rule above — are studied precisely because they answer a different question. The nucleolus is efficient, symmetric and respects null players, and is the split whose loudest objection is quietest rather than a linear function of the game, and it exists whether or not the core does.

Efficiency is essentially never dropped, because a rule that does not divide up what there is answers no question anybody asked — the same reason a split nobody can walk away from is defined only on splits that add up.

The null-player condition is never dropped either, for the same reason: a rule paying somebody for nothing has lost the connection to the game.

The same pattern holds in the neighbouring anchors: no stable matching rule is safe from a lie because one condition there cannot be dropped either, and the literature is a survey of which of the others can. So the practical reading of the independence result is that it maps the neighbourhood. Two of the conditions are structural and two are choices, and each choice has a family of alternatives on the other side of it.

The general shape of an independence proof #

The manoeuvre here is standard across mathematics and is worth naming, because it is what distinguishes a characterisation from a list.

To show that a list of axioms is independent, exhibit for each axiom a structure satisfying all the others and failing that one. The classical instance is the parallel postulate, where a model of the other axioms in which it fails settles two thousand years of attempts to derive it.

The two settings differ in what the exhibit is. In geometry it is a model — a world where the axioms hold. Here it is a rule — a function from games to splits. In both cases the exhibit is the whole proof, and in both cases producing it is the entire difficulty: nothing about the axioms tells anybody where to look.

A list of axioms with no independence proof is a list that might be redundant, and redundancy is not a harmless inefficiency. An axiom that follows from the others is one that cannot be dropped to weaken the theory, so a characterisation with a passenger describes a smaller class of alternatives than it appears to.

Other characterisations of the same rule #

A rule with one axiomatic characterisation usually has several, and the Shapley value has an unusual number. They are worth listing because each replaces the awkward condition with a different awkward condition, and the choice between them is a choice about what to find intuitive.

Balanced contributions. Replace additivity and symmetry by: for any two players, what the first loses when the second leaves equals what the second loses when the first leaves. That single condition plus efficiency gives the same rule, and it is much easier to defend as fair — it says the relationship between any two players is symmetric.

The potential function. There is a single function on games whose discrete derivative in each player’s direction is that player’s share, and efficiency is the statement that the derivatives sum to the game’s value. That characterisation replaces four conditions with one object and is the route most computational work takes.

Consistency. If a player leaves with their share, the rule applied to the reduced game among the rest gives the rest the same shares as before. Efficiency, symmetry and consistency characterise the rule, with no additivity anywhere — which matters because additivity is the condition nobody finds compelling.

The three lists share efficiency and share nothing else that is not implied. That a single rule is picked out by three unrelated specifications is the strongest argument for it there is — much stronger than any one of the specifications, none of which is obviously the right thing to want.

There is a general point in that, and it is the reason a field bothers to reprove a theorem it already has. A characterisation is only as convincing as its least convincing condition, and additivity is a technical requirement about combining unrelated ventures rather than a statement about fairness. Finding a second route that avoids it does not make the rule more true; it makes the reason to use it independent of a condition somebody might reasonably reject.

What the picture cannot show #

The figure decides its ticks and crosses on one game, and the conditions are statements about all games. A rule failing a condition on one game fails it, which is what the crosses mean and is sound. A rule satisfying a condition on one game does not satisfy it in general, which is what the ticks appear to mean and do not.

That gap is real and is the figure’s honest limitation. The four rules do satisfy their three conditions in general, and the argument for each is two lines of algebra given in the prose above — linearity for the halved value, telescoping for the marginal vector, symmetry of the total for the equal split. The picture reports an instance and the prose carries the proof.

It also cannot show the uniqueness theorem, which is what the columns are independence proofs for. That the four conditions together admit exactly one rule is a decomposition argument over a basis of $2^n - 1$ games, and there is no picture of a basis of that size.

Where the ladder goes next #

Above: the rule applied where the game is a vote, and the discovery that a share of the votes and a share of the power are different numbers. Then what to do when there are too many orders to average over. Then the same rule dividing a bill instead of a surplus.

One debt. The weighted Shapley values are named here as the family obtained by dropping symmetry and are not drawn. They have their own characterisation — the same four conditions with symmetry replaced by a consistency requirement — and the family is a genuinely useful object rather than a footnote.

What the second half was for #

A characterisation needs both halves: the conditions are enough, and each of them is needed.

The second half is the one that says the list was well chosen, and it is proved by exhibiting a rule for each condition that keeps the others and breaks it. Doing that here turns the four conditions from a set of demands into a map: two of them are structural and cannot sensibly be dropped, and two of them are choices with families of alternatives waiting on the other side.